Role of nonlinear localized structures and turbulence in magnetized plasma

Abstract

In the present study, we have analyzed the field localization of kinetic Alfvén wave (KAW) due to the presence of background density perturbation, which are assumed to be originated by the three dimensionally propagating low frequency KAW. These localized structures play an important role for energy transportation at smaller scales in the dispersion range of magnetic power spectrum. For the present model, governing dynamic equations of high frequency pump KAW and low frequency KAW has been derived by considering ponderomotive nonlinearity. Further, these coupled equations have been numerically solved to analyze the resulting localized structures of pump KAW and magnetic power spectrum in the magnetopause regime. Numerically calculated spectrum exhibits inertial range having spectral index of $-3/2$ followed by steeper scaling; this steepening in the turbulent spectrum is a signature of energy transportation from larger to smaller scales. In this way, the proposed mechanism, which is based on nonlinear wave-wave interaction, may be useful for understanding the particle acceleration and turbulence in magnetopause.

Appendix: Expressions for ponderomotive force

For electrons, the components of ponderomotive force due to 3-D KAW are

$$\begin{aligned} F_{ex} &= - \frac{m_{e}}{4} \biggl[ \biggl( \frac{c\alpha k_{0y}}{B_{0}} \biggr) ^{2} - \biggl( \frac{k_{0 \bot }^{2}}{4\pi n_{0}m_{i}} \biggr) \biggr] \frac{ \partial \vert A_{z} \vert ^{2}}{\partial x} \\ &\quad + \frac{m_{e}}{4} \biggl( \frac{k_{0x}k_{0y}c^{2}\alpha^{2}}{B_{0}^{2}} \biggr) \frac{\partial \vert A_{z} \vert ^{2}}{\partial y} \\ &\quad - \frac{m_{e}}{4} \biggl[ \biggl( \frac{ic ^{2}\alpha k_{0y}k_{0 \bot }^{2}}{4\pi n_{0}eB_{0}} \biggr) \biggr] \frac{ \partial \vert A_{z} \vert ^{2}}{\partial z} \\ F_{ey} &= \frac{m_{e}}{4} \biggl( \frac{c\alpha k_{0y}}{B_{0}} \biggr) ^{2}\frac{\partial \vert A_{z} \vert ^{2}}{\partial x} - \frac{m _{e}}{4} \biggl[ \biggl( \frac{c\alpha k_{0x}}{B_{0}} \biggr) ^{2} \\ &\quad - \biggl( \frac{k_{0 \bot }^{2}}{4\pi n_{0}m_{i}} \biggr) \biggr] \frac{ \partial \vert A_{z} \vert ^{2}}{\partial y} + \frac{m_{e}}{4} \biggl[ \biggl( \frac{ic^{2}\alpha k_{0y}}{4\pi n_{0}eB_{0}} \biggr) \biggr] \frac{ \partial \vert A_{z} \vert ^{2}}{\partial z} \\ F_{ez} &= \frac{m_{e}}{4} \biggl[ \biggl( \frac{i\alpha k_{0y}k_{0 \bot } ^{2}c^{2}}{4\pi n_{0}eB_{0}} \biggr) + \biggl( \frac{i\alpha ek_{0y}}{m _{e}B_{0}} \biggr) \biggr] \frac{\partial \vert A_{z} \vert ^{2}}{ \partial x} \\ &\quad - \frac{m_{e}}{4} \biggl[ \biggl( \frac{i\alpha k_{0x}k_{0 \bot }^{2}c^{2}}{4\pi n_{0}eB_{0}} \biggr) + \biggl( \frac{i\alpha ek _{0x}}{m_{e}B_{0}} \biggr) \biggr] \frac{\partial \vert A_{z} \vert ^{2}}{\partial y} \\ &\quad - \frac{m_{e}}{4} \biggl[ \biggl( \frac{k_{0 \bot } ^{2}c}{4\pi n_{0}e} \biggr) ^{2} \biggr] \frac{\partial \vert A_{z} \vert ^{2}}{\partial z} \end{aligned}$$

Similarly for ions, the components of ponderomotive force due to 3-D KAW are

$$\begin{aligned} F_{ix} &= - \frac{m_{i}}{4} \biggl[ \biggl( \frac{Pc\alpha k_{0y}}{B_{0}} \biggr) ^{2} + \biggl( \frac{Qc\alpha k_{0x}}{B_{0}} \biggr) ^{2} \\ &\quad - \frac{e ^{2}}{m_{i}^{2}} \biggl( \frac{k_{0z}\alpha }{\omega_{0}c} - \frac{1}{c ^{2}} \biggr) \biggr] \frac{\partial \vert A_{z} \vert ^{2}}{\partial x} \\ &\quad - \frac{m_{i}}{4} \biggl[ \biggl( \biggl( \frac{Qc\alpha }{B_{0}} \biggr) ^{2} - \biggl( \frac{Pc\alpha }{B_{0}} \biggr) ^{2} \biggr) k_{0x}k_{0y} \\ &\quad + i \biggl( \frac{c\alpha k_{0x}}{B_{0}} \biggr) ^{2}PQ + i \biggl( \frac{c \alpha k_{0Y}}{B_{0}} \biggr) ^{2}PQ \biggr] \frac{\partial \vert A _{z} \vert ^{2}}{\partial y} \\ &\quad - \frac{m_{i}}{4} \biggl[ \biggl( \frac{Qe\alpha k_{0x}}{m_{i}B_{0}} \biggr) + i \biggl( \frac{Pe\alpha k_{0y}}{m_{i}B_{0}} \biggr) \\ &\quad - i \biggl( \frac{ceP \alpha^{2}}{m_{i}B_{0}\omega_{0}} \biggr) k_{0Y}k_{0z} - \biggl( \frac{Qce \alpha^{2}}{m_{i}B_{0}\omega_{0}} \biggr) k_{0x}k_{0z} \biggr] \frac{ \partial \vert A_{z} \vert ^{2}}{\partial z} \\ F_{iy} &= - \frac{m_{i}}{4} \biggl[ \biggl( \biggl( \frac{Qc\alpha }{B_{0}} \biggr) ^{2} - \biggl( \frac{Pc\alpha }{B_{0}} \biggr) ^{2} \biggr) k_{0x}k_{0y} \\ &\quad - i \biggl( \frac{c\alpha k_{0y}}{B_{0}} \biggr) ^{2}PQ - i \biggl( \frac{c\alpha k_{0x}}{B_{0}} \biggr) ^{2}PQ \biggr] \frac{ \partial \vert A_{z} \vert ^{2}}{\partial x} \\ &\quad - \frac{m_{i}}{4} \biggl[ \biggl( \frac{Pc\alpha k_{0x}}{B_{0}} \biggr) ^{2} + \biggl( \frac{Qc\alpha k_{0y}}{B_{0}} \biggr) ^{2} \\ &\quad - \frac{e^{2}}{m _{i}^{2}} \biggl( \frac{k_{0z}\alpha }{\omega_{0}c} - \frac{1}{c^{2}} \biggr) \biggr] \frac{\partial \vert A_{z} \vert ^{2}}{\partial y} \\ &\quad - \frac{m_{i}}{4} \biggl[ i \biggl( \frac{Pce\alpha^{2}}{m_{i}B_{0}\omega _{0}} \biggr) k_{0x}k_{0z} - \biggl( \frac{Pce\alpha^{2}}{m_{i}B_{0} \omega_{0}} \biggr) k_{0y}k_{0z} \\ &\quad - i \biggl( \frac{Pe\alpha k_{0x}}{m _{i}B_{0}} \biggr) + \biggl( \frac{Qe\alpha k_{0y}}{m_{i}B_{0}} \biggr) \biggr] \frac{\partial \vert A_{z} \vert ^{2}}{\partial z} \\ F_{iz}& = - \frac{m_{i}}{4} \biggl[ i \biggl( \frac{Pce\alpha^{2}}{m_{i}B _{0}\omega_{0}} \biggr) k_{0y}k_{0z} - \biggl( \frac{ceQ\alpha^{2}}{m _{i}B_{0}\omega_{0}} \biggr) k_{0x}k_{0z} \biggr] \\ &\quad \times \frac{\partial \vert A_{z} \vert ^{2}}{\partial x} - \frac{m_{i}}{4} \biggl[ i \biggl( \frac{Pce\alpha^{2}}{m_{i}B_{0}\omega _{0}} \biggr) k_{0x}k_{0z} \\ &\quad + \biggl( \frac{ceQ\alpha^{2}}{m_{i}B_{0} \omega_{0}} \biggr) k_{0y}k_{0z} \biggr] \frac{\partial \vert A_{z} \vert ^{2}}{\partial y} - \frac{m_{i}}{4} \biggl[ \biggl( \frac{ek_{0z}\alpha }{m_{i}\omega_{0}} \biggr) ^{2} \\ &\quad - 2 \biggl( \frac{e^{2}k_{0z}\alpha }{m_{i}^{2}\omega_{0}c} \biggr) + \biggl( \frac{e}{m_{i}c} \biggr) ^{2} \biggr] \frac{\partial \vert A _{z} \vert ^{2}}{\partial z} \end{aligned}$$

Here $P = \frac{\omega_{ci}^{2}}{ ( \omega_{ci}^{2} - \omega_{0} ^{2} ) }$, $Q = \frac{\omega_{ci}\omega_{0}}{ ( \omega_{ci} ^{2} - \omega_{0}^{2} ) }$, $\alpha = \frac{v_{A}^{2}k_{0z}}{c \omega_{0}}$.